Average Error: 61.5 → 2.3
Time: 15.5s
Precision: binary64
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]
↓
\[4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi} + -0.08333333333333333 \cdot \left(\pi \cdot \left(f \cdot f\right)\right)\]
-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)
↓
4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi} + -0.08333333333333333 \cdot \left(\pi \cdot \left(f \cdot f\right)\right)
(FPCore (f)
 :precision binary64
 (-
  (*
   (/ 1.0 (/ PI 4.0))
   (log
    (/
     (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f))))
     (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))))))
↓
(FPCore (f)
 :precision binary64
 (+
  (* 4.0 (/ (- (log f) (log (/ 4.0 PI))) PI))
  (* -0.08333333333333333 (* PI (* f f)))))
double code(double f) {
	return -((1.0 / (((double) M_PI) / 4.0)) * log((exp((((double) M_PI) / 4.0) * f) + exp(-((((double) M_PI) / 4.0) * f))) / (exp((((double) M_PI) / 4.0) * f) - exp(-((((double) M_PI) / 4.0) * f)))));
}
↓
double code(double f) {
	return (4.0 * ((log(f) - log(4.0 / ((double) M_PI))) / ((double) M_PI))) + (-0.08333333333333333 * (((double) M_PI) * (f * f)));
}

Error

Bits error versus f

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 61.5

    \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\]

  2. Simplified61.5

    \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{e^{\frac{\pi}{4} \cdot f} - {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}\right) \cdot \frac{-4}{\pi}}\]

  3. Taylor expanded around 0 2.4

    \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\color{blue}{\left(0.03125 \cdot \left({\pi}^{2} \cdot {f}^{2}\right) + \left(0.0026041666666666665 \cdot \left({f}^{3} \cdot {\pi}^{3}\right) + 0.25 \cdot \left(\pi \cdot f\right)\right)\right) - \left(0.16666666666666666 \cdot \left({f}^{3} \cdot \left({\log \left(e^{-0.25}\right)}^{3} \cdot {\pi}^{3}\right)\right) + \left(f \cdot \left(\log \left(e^{-0.25}\right) \cdot \pi\right) + 0.5 \cdot \left({\pi}^{2} \cdot \left({\log \left(e^{-0.25}\right)}^{2} \cdot {f}^{2}\right)\right)\right)\right)}}\right) \cdot \frac{-4}{\pi}\]

  4. Simplified2.4

    \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + {\left(e^{-0.25}\right)}^{\left(\pi \cdot f\right)}}{\color{blue}{{\left(\pi \cdot f\right)}^{3} \cdot 0.0026041666666666665 + \left(\left(\pi \cdot f\right) \cdot \left(0.25 + \left(\pi \cdot f\right) \cdot 0.03125\right) + \left({\left(\frac{\pi}{4} \cdot f\right)}^{3} \cdot 0.16666666666666666 - \left(f \cdot \left(\pi \cdot -0.25\right) + 0.5 \cdot \left({\pi}^{2} \cdot \left(\left(f \cdot f\right) \cdot 0.0625\right)\right)\right)\right)\right)}}\right) \cdot \frac{-4}{\pi}\]

  5. Taylor expanded around 0 2.3

    \[\leadsto \color{blue}{\left(0.010416666666666666 \cdot \left({f}^{2} \cdot \pi\right) + \left(0.25 \cdot \left({f}^{2} \cdot \left(\log \left(e^{-0.25}\right) \cdot \pi\right)\right) + 4 \cdot \frac{\log f}{\pi}\right)\right) - \left(0.5 \cdot \left({f}^{2} \cdot \left({\log \left(e^{-0.25}\right)}^{2} \cdot \pi\right)\right) + \left(2 \cdot \left(f \cdot \log \left(e^{-0.25}\right)\right) + \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + 0.5 \cdot f\right)\right)\right)}\]

  6. Simplified2.3

    \[\leadsto \color{blue}{\left(0.010416666666666666 \cdot \left(\pi \cdot \left(f \cdot f\right)\right) + \left(0.25 \cdot \left(f \cdot \left(f \cdot \left(\pi \cdot -0.25\right)\right)\right) + 4 \cdot \frac{\log f}{\pi}\right)\right) - \left(0.5 \cdot \left(\pi \cdot \left(\left(f \cdot f\right) \cdot 0.0625\right)\right) + \left(f \cdot -0.5 + \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + f \cdot 0.5\right)\right)\right)}\]

  7. Taylor expanded around 0 2.3

    \[\leadsto \color{blue}{4 \cdot \frac{\log f}{\pi} - \left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right)}{\pi} + 0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)\right)}\]

  8. Simplified2.3

    \[\leadsto \color{blue}{4 \cdot \left(\frac{\log f}{\pi} - \frac{\log \left(\frac{4}{\pi}\right)}{\pi}\right) + -0.08333333333333333 \cdot \left(\pi \cdot \left(f \cdot f\right)\right)}\]
  9. Using strategy rm

  10. Applied sub-div_binary642.3

    \[\leadsto 4 \cdot \color{blue}{\frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi}} + -0.08333333333333333 \cdot \left(\pi \cdot \left(f \cdot f\right)\right)\]

  11. Final simplification2.3

    \[\leadsto 4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi} + -0.08333333333333333 \cdot \left(\pi \cdot \left(f \cdot f\right)\right)\]

Reproduce

herbie shell --seed 2020292 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary64
  (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))) (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))))))